This just occurred to me. I am so furious at these people, but glad that I decided not to let them 'transfer me' to the FBI and tried calling Citi back myself.
Step-by-step what they did:
I got a call from a lady claiming to be from the Citibank Fraud Prevention Department. She even gave me a fake ID number 'CTIxxxx' and gave me the real number of Citi's credit card line (800-950-5114). She told me my card was used to buy a gun in New York. She sounded professional and concerned, not aggressive.
She gave me a ton of fake and true details. True: Citi contact number, a real Citi address in Manhattan. Fake (almost certainly): a Citi credit card number; a case number; a card mailing address.
They then offered to 'transfer' me to someone at the NY FBI. Fortunately, when I asked them to also give me the FBI number, they didn't provide it but said they could transfer directly. My BS detector finally went up and I said I'd call them myself.
My first follow-up call was to Citi directly. Unfortunately it's harder to talk to a human at Citi if you don't actually have a known account number and don't want to punch in your SSN. I had to try a few different times of button mashing and finally got to a human who confirmed the card number didn't exist. I also called the actual number of the branch address they gave me, and the rep confirmed I didn't have my account in their records under my phone number.
Although I am lucky to not know anyone personally who has fallen for this, I am furious that this is basically a business of lying and manipulating people. Small pleasures like watching the Beekeeper pale in comparison to the real harm that happens all the time. Be careful out there folks, don't trust anyone over the phone. I probably should have just ignored the unrecognized number with the familiar area code.